In the framework of a national research project, “ESSOR” [ess09], in collab-oration with L2S research team, transmissions over noisy channels have been studied, and in particular multiple description video coding. MDC without side information taken into account for the time being is considered here. A focus is done on the transmission of several descriptions over noisy channels.
The crucial problem of this kind of schemes is the optimal decoding of the source, based on the noisy received descriptions. Two decoding algorithms are proposed here. A first one tries to estimate the two generated descrip-tions from the received channel outputs. A second focuses on the direct estimation of the source from the two noisy descriptions, without trying to estimate the single descriptions. Simulation results show a good robustness of the proposed decoding schemes against transmission errors.
6.1 Structure of the considered multiple description coder The approaches of decoding of the source presented in the following may be applied to many MD coding schemes. Though, the MD coding scheme considered here is based on the previous video coder presented in Section 3.1, and thus performs a motion-compensated spatio-temporal DWT of the video frames. Redundancy is introduced before quantization, the balanced descriptions being produced thanks to a bit allocation based on the charac-teristics of the channel.
6.1.1 General structure
The joint source channel (JSC) coding scheme used here corresponds to the class of JSC where the redundancy is introduced before source coding as proposed in [GKD07] (see the Figure 6.1). Here, the joint encoder consists in (see the Figure 6.2):
- a spatio-temporal DWT of the source data used to generate the dif-ferent balanced descriptions by duplication (the wavelet coefficients are
109
Figure 6.1: Proposed JSC coding scheme where the redundancy is intro-duced on the quantized wavelet coefficients during the bit allocation step (joint encoding box).
repeated in both the descriptions), as in [PAB03b]. In this case, the source pdf can be modeled by a generalized Gaussian [PAB03a];
- a model-based bit allocation, which dispatches the bits across the differ-ent subbands and the differdiffer-ent descriptions, according to an information on the channel noise, and thus on the needed redundancy [PAB03b]. No predictive feedback is used;
- a scalar or vector quantization, followed by a fixed-length coding.
6.1.2 The bit allocation
The considered MDC scheme, based on the thesis work of M. Pereira [PAB03b], focus on the special case where a transmitter and a receiver are linked by two channels of equal capacity. Thus, this MDC scheme is a balanced MDC (BMDC). A BMDC framework generates descriptions of equal rate and im-portance. Explicit redundancy is introduced, so that each wavelet coefficient is transmitted more than once and coded with a different accuracy each time.
The DWT is performed and then, the wavelet coefficients are repeated in both the descriptions. When a subband is finely coded in one description, the algorithm forces it to be coarsely coded in the other, as seen in Figure 6.3.
6.1.2.1 The bit allocation problem
The problem of the bit allocation is thus to find, for a given redundancy be-tween the descriptions, the combination of scalar quantizers [SG88, Ort00]
across the various wavelet coefficients subbands that will produce the mini-mum total central distortion D0, while satisfying constraints on the side bit rates R1 and R2, and on the side distortions D1 and D2. This problem can
6.1. Structure of the considered multiple description coder 111
Figure 6.2: General coding scheme.
be resumed as:
with RT the target output bit rate and Dm the maximal side distortion imposed, and quanti-zation step and σi,j the variance of the i-th subband. The parameter ai in (6.1) is the size of the i-th subband divided by the size of the sequence, and Ri,j(˜qi,j) is the output bit rate in bits per sample for the i-th subband.
with Di,j(˜qi,j) the quantization distortion of the i-th subband of description j, ∆i an optional weight for frequency selection and wi the weights of the
filter bank [B.U96] for the i-th subband.
Let us define the central distortion for the i-th subband as (see [Per04]):
Di,0(˜qi,1, ˜qi,2) = 1 where ρN is a weighting parameter, called the redundancy parameter which returns an information on the channel noise. The redundancy parameter domain is [0, 1], ρN = 0 is used when the channel is noiseless, and ρN = 1 is used when a very noisy channel is expected. The amount of redundancy, i.e., the importance of the redundant subbands, have to depend on the channel model. It determines the intermediate redundancies, and implicitly the intermediate values of the ρN parameter. An example of computation of this parameter will be presented in Section 6.2.2.2.
6.1.2.2 The functional to optimize
The Lagrangian functional J for the constrained optimization problem is then given by: with λj and µj some Lagrangian multipliers, and
D0 = XN
i=1
∆iwiσ2i,0Di,0(˜qi,1, ˜qi,2) ,
with σi,0 the variance of the i-th subband of the central description, and Di,0(˜qi,1, ˜qi,2) the central distortion of the i-th subband.
The constraint Fj on the bit rate can be expressed as, for the different descriptions j = 1, 2:
Fj = Because balanced descriptions are wanted, a penalty on the side distortions is needed. By considering a constraint x < 0, the penalty can be written as:
P (x) =
6.1. Structure of the considered multiple description coder 113
Figure 6.3: Example of division of the wavelet subbands between primary subbands (finely coded) and redundant subbands (coarsely coded) in the two descriptions.
The penalty Pj can thus be expressed as:
Pj =
|Dj(Rj) − DM| + (Dj(Rj) − DM) 2
2
, ∀j ∈ {1, 2}. (6.8) Considering the central distortion given by (6.3), the bit rate constraint (6.5), and the side distortion penalty (6.8), the Lagrangian functional (6.4) can be expressed as:
6.1.2.3 The solution
Finally, the resolution of the following system (with first order conditions) gives the optimal sets of quantization steps {˜qi,1}, {˜qi,2}, for a given ρN, and With the Ci,j parameter defined as:
Ci,j =
1, if min(σ2i,1Di,1(˜qi,1), σ2i,2Di,2(˜qi,1)) = σ2i,jDi,j(˜qi,j), ∀j ∈ {1, 2}
ρN, otherwise.
and the Ej parameter computed from:
Ej =
2 × (Dj(Rj) − DM) , if Dj(Rj) > DM, ∀j ∈ {1, 2}
0 otherwise.
More details can be found in [PAB03b] or [Per04].
6.2 Multiple description decoding
In this thesis, contrary to the work of M. Pereira, the focus is not done in the coder side of the MDC scheme, but on the decoding part. As said in Section 5.7, the decoding of descriptions transmitted over error prone channels is a crucial problem and has not been so much explored in video transmission.
In order to optimally decode the central description, two approaches have been implemented: the first one tries to construct the central description by first evaluating the two side descriptions from the received channel out-puts, whereas the second one focuses on the direct estimation of the central description from the two noisy side descriptions, without trying to estimate these descriptions.
6.2. Multiple description decoding 115
Noise
Noise
MDC Decoding
s1
s2
r1
r2
sb0
Figure 6.4: General scheme of the decoding of two noisy descriptions.
6.2.1 Problem statement
At the decoder, the challenge is to reconstruct a ‘central’ signal with central distortion D0 as small as possible, using the knowledge of the two side descriptions. Figure 6.4 illustrates the problem. s1 and s2 represent the quantized source descriptions at emitter (quantization values issued from an optimal codebook C of M symbols before binary coding), and r1 and r2 represent the observed noisy descriptions at the receiver. The signal bs0 is the reconstructed signal obtained by decoding the two received descriptions (here with the system described in Section 6.1.2). In the following sections, two algorithms for optimal decoding are described.
6.2.2 Decoding using a Model-Based MAP and a decision ap-proach
The problem of choosing the best description at decoder (resumed in Figure 6.5), for each symbol (i.e. here the quantized wavelet coefficients), can be seen as a maximum a posteriori (MAP) estimation problem, consisting in determining:
(s∗1, s∗2) = arg max
s0
p(s1, s2|r1, r2), or equivalently by:
(s∗1, s∗2) = arg min
s1,s2 − log [p(s1, s2|r1, r2)] . Let recall that (according to Bayes’ rule):
p(s|r) = p(s)p(r|s) p(r) , or even:
p(s|r) ∼ p(s)p(r|s),
where ∼ stands for proportional to, and since p(r), the density of the ob-served value, is not taken into account in the optimization. In an equivalent way, the joint conditional density can be written as:
p(s1, s2|r1, r2) ∼ p(s1, s2|r2)p(r1|s1, s2, r2)
Decision MAP
s∗1 s∗2 r1
r2
b s0
Figure 6.5: First proposed decoding approach: model-based MAP and de-cision approach.
Moreover,
p(r1|s1, s2, r2) = p(r1|s1),
since r1, knowing s1, can be supposed to not directly depend on the value of s2 or the value of r2. Thus,
p(s1, s2|r1, r2) ∼ p(s1, s2|r2)p(r1|s1)
∼ p(s1, s2)p(r2|s1, s2)p(r1|s1)
∼ p(s1, s2)p(r1|s1)p(r2|s2).
The criterion to minimize is then given by:
(s∗1, s∗2) = argmin
s1,s2
− log [p(s1, s2)p(r1|s1)p(r2|s2)] . (6.9) The minimization of this criterion provides two optimal values s∗1 and s∗2, for each coefficient, and two distortions called D1 and D2 with, classically:
Di=X
j∈C
p(ri|sj) Z s∗i+q
2
s∗i−q2 (x − s∗i)2pS(x)dx,
where q is the quantization step, and pS(x) is the probability density func-tion (pdf) of the source signal (i.e. the wavelet subbands) from which the two descriptions s1 and s2 are obtained. Note that in the case where the noise is introduced by a channel assumed to be memoryless, P
j∈Cp(ri|sj) corresponds to the sum of the transition probabilities over all the possible inputs. In that case, Di corresponds to the distortion introduced by the source quantizer and the channel.
Then, bs0, the optimal reconstruction value at decoding, is set to s∗1 or s∗2 according to the values of D1 and D2 and using the following rule:
bs0= s∗1 if min(D1, D2) = D1 b
s0= s∗2 else if.
The evaluation of the term p(s1, s2) is presented in what follows.
6.2. Multiple description decoding 117
Figure 6.6: Interval containing the source data quantized by s1in description 1 and by s2 in description 2.
6.2.2.1 Evaluation of p(s1, s2)
In the case of uniform scalar quantization, it is possible to give an expression of p(s1, s2) in function of the quantization steps and of the source pdf. As it can be seen in Figure 6.6, this density can be evaluated as:
p(s1, s2) =
Z min(s1+q12 ,s2+q22) max(s1−q12 ,s2−q22)
pS(x)dx with (s1, s2) ∈ C1× C2,
where Ci is the quantization codebook for the description i.
This relation holds if max(s1−q21, s2−q22) is smaller than min(s1+q21, s2+q22).
If it is not the case, the value of p(s1, s2) is set to zero.
Since the function p(s1, s2) depends on a ‘min’ and a ‘max’, four different cases have to be considered. Let first define the quantity ∆S = s1− s2.
• Case 1: max(s1−q21, s2−q22) = s1−q21 and min(s1+q21, s2+q22) = s1+q21. That is to say ∆S ≥ 12(q1− q2) and ∆S < 12(q2− q1). These conditions hold for q2 ≥ q1. Then:
p(s1, s2) =
Z s1+q1
2
s1−q12
pS(x)dx,
• Case 2: max(s1−q21, s2−q22) = s1−q21 and min(s1+q21, s2+q22) = s2+q22. That is to say ∆S ≥ 12(q1− q2) and ∆S > 12(q2− q1). These conditions hold for all q1 and q2. Then:
p(s1, s2) =
Z s2+q2
2
s1−q12
pS(x)dx,
• Case 3: max(s1−q21, s2−q22) = s2−q22 and min(s1+q21, s2+q22) = s1+q21. That is to say ∆S ≤ 12(q1− q2) and ∆S< 12(q2− q1). These conditions hold for all q1 and q2. Then:
p(s1, s2) =
Z s1+q12
s2−q22
pS(x)dx,
• Case 4: max(s1−q21, s2−q22) = s2−q22 and min(s1+q21, s2+q22) = s2+q22. That is to say ∆S ≤ 12(q1− q2) and ∆S> 12(q2− q1). These conditions hold for q1 ≥ q2. Then:
p(s1, s2) =
Z s2+q2
2
s2−q22
pS(x)dx,
6.2.2.2 Channel model
To compute (6.9), one has to evaluate p (ri|si), i = 1, 2. Let define u (s) as a function which represents the source s after quantization, fixed-length M -bit binary indexation, and BPSK signalling. Let {−1, 1}M be the set of all values which may be taken by u (s). Then one has:
p (ri|si) = X
u∈{−1,1}M
p (ri, u|si)
= X
u∈{−1,1}M
p (ri|u, si) p (u|si) . (6.10)
In (6.10), p (u|si) is directly determined from the quantization, indexation, and modulation of si, i.e.,
p (u|si) =
1 if u = u (si) 0 else.
Thus, (6.10) simplifies to:
p (ri|si) = p (ri|u (si) , si) .
For what concerns the channel output, u (si) provides as much information on ri as si does (si → u (si) → ri forms a Markov chain), thus, one finally gets:
p (ri|si) = p (ri|u (si)) . (6.11) Assuming that the channel is zero-mean white Gaussian with noise variance σ2, (6.11) becomes:
p (ri|si) = 2πσ2−M/2
exp −|ri− u (si)|2 2σ2
!
. (6.12)
6.2. Multiple description decoding 119
2 3 4 5 6 7 8
20 22 24 26 28 30 32 34 36
SNR (dB)
PSNR (dB)
Central (proposed) Side
Figure 6.7: First approach: PSNR comparisons for Foreman between the side noisy description and the central description, bit-rate Rt= 2 Mbps.
It can be noted that, for such a channel, the redundancy parameter ρN is computed as:
ρN = 1 −B log2(1 +NS)
2 ,
with B the channel bandwidth in symbol/s, and NS the SNR (where S is the received signal power and N is the AWGN power within the channel bandwidth). The details of computation for this expression can be found in [Per04].
Of course, other types of channel could also be considered, but the focus here is only done on this special case.
6.2.2.3 Experimental results of the first approach
The experiments have been done in different conditions, on the CIF se-quences Foreman and Erik, and on the SD sequence city, with 3 temporal decomposition levels, and with quarter-pixelic motion vectors. Each wavelet coefficient is quantized with a scalar quantization, encoded, and transmit-ted using BPSK signalling over an AWGN channel, with a noise variance σ2. Headers and motion information are assumed noise free.
Figure 6.7 presents some PSNR comparisons between the side noisy de-scription and the central dede-scription obtained with the proposed algorithm, for Foreman. PSNR values are presented for a bit-rate of 2 Mbps, and for different SNR values, induced by the channel noise. Figures 6.8(a) and 6.8(b) present the same kind of results for the sequences Erik (Rt = 2
2 3 4 5 6 7 20
22 24 26 28 30 32
SNR (dB)
PSNR (dB)
Central (proposed) Side
(a) erik.
2 3 4 5 6 7
21 22 23 24 25 26 27 28 29
SNR (dB)
PSNR (dB)
Central (proposed) Side
(b) city.
Figure 6.8: First approach: PSNR comparisons for erik (Rt= 2 Mbps) and city (Rt = 2.5 Mbps) between the side noisy description and the central description.
6.2. Multiple description decoding 121
Foreman Foreman
1.5 Mbps Side Central 2 Mbps Side Central SN R = 3 20.67 30.34 SN R = 3 20.98 33.50 SN R = 7 23.74 33.72 SN R = 7 26.84 35.6
Erik Erik
1.5 Mbps Side Central 2 Mbps Side Central SN R = 2 19.03 28.23 SN R = 2 19.31 29.31 SN R = 6 24.15 31.32 SN R = 6 22.92 32.67
City City
2.5 Mbps Side Central 2.8 Mbps Side Central SN R = 4 23.7 27.68 SN R = 4 24.12 28.21 SN R = 7 25.68 28.28 SN R = 7 25.88 28.35
Table 6.1: First approach: PSNR (dB) comparisons between the side de-scription and the central dede-scription obtained with the first approach; for the sequences Foreman, Erik and city (on three (2,0) decomposition levels, with quarter-pixel motion vectors), for different bit-rates, and with different values of SN R.
Mbps) and city (Rt = 2.5 Mbps). The gain in PSNR is very important:
the performances in respect with the noisy side description can be improved up to 11 dB.
Some interesting visual results are then presented. Figure 6.9(a) shows, for a SNR equal at 3 dB, reconstructed images of Foreman, at Rt = 2 Mbps, for the noiseless images, the noisy side description and for the central description, obtained by applying the proposed MAP decoding algorithm.
Figures 6.9(b) and 6.9(c) show the same results for Erik (SN R = 4 dB, Rt = 2 Mbps) and city (SN R = 4 dB, Rt = 2.5 Mbps). The central de-scriptions (j) are better preserved, thanks to the proposed algorithm. The noiseless images are almost retrieved.
In Table 6.1 are also summarized the PSNR comparisons between the side description and the central description obtained by applying the first approach, for different values of SN R, with the same coding parameters than previously for Foreman, Erik and city. The results are really good, the PSNR of the central description is always widely higher than the one of the side description.
6.2.3 Decoding by a direct estimation of the central description A different method of decoding can be considered, based on a direct evalu-ation of the value bs0 of the central description.
(i) (j) (k)
(a) Foreman, SN R = 3 dB, Rt= 2 Mbps, images 13 and 117.
(b) erik, SN R = 4 dB, Rt= 2 Mbps, images 13 and 44.
(c) city, SN R = 4 dB, Rt= 2.5 Mbps, images 18 and 44.
Figure 6.9: First approach: visual results, (i) noiseless images, (j) central description, (k) side description.
6.2. Multiple description decoding 123
Figure 6.10: Second proposed decoding approach: direct estimation of the central description.
6.2.3.1 Estimation of the solution
The estimate bs0 of the central description is obtained from the probability of s0 knowing the channel outputs r1 and r2 (see Figure 6.10):
bs0= arg max
s0
p (s0|r1, r2) , (6.13) and can be expressed as:
bs0 = arg max
s0
X
s1,s2
p (s0, s1, s2|r1, r2) .
Using Bayes’ rule, one can have:
bs0 = arg max and finally, since the two channels are independent:
b p(s0) is the pdf of the source (i.e. in the case of wavelet subbands, a gen-eralized Gaussian [PAB03a]). p(ri/si) are the transition probabilities of the channel. p(s1, s2/s0) is computed thanks to the values of quantization of the side descriptions. More precisely:
p(s1, s2/s0) = 1 if s0 ∈ P1∩ P2
p(s1, s2/s0) = 0 if not, (6.15) with P1 and P2 the quantization intervals associated to s1 and s2.
6.2.3.2 Channel model
In the same channel conditions as in the previous sections, the same expres-sion as in 6.2.2.2 is used for the channel model:
p (ri|u (si)) = 2πσ2−M/2
exp −|ri− u (si)|2 2σ2
! .
Then, this equation and (6.15) may be combined in (6.14) to get the cost function
J (s0) = p(s0)X
s1,s2
exp −|r1− u (s1)|2+ |r2− u (s2)|2 2σ2
!
.p (s1, s2|s0)
, (6.16)
which maximization leads to the estimation of s0.
As si is totally determined by s0 and by the quantization step chosen for the description i, one can write: si = Qi(s0), and thus:
J (s0) = exp −|r1− u (Q1(s0))|2+ |r2− u (Q2(s0))|2 2σ2
!
.p Q1(s0), Q2(s0)|s0
.p(s0).
6.2.3.3 Experimental results of the second approach
As for the first approach, the experiments have been done in different con-ditions, using a Gaussian channel with a noise variance σ2 in order to add noise at the 2 descriptions, on the CIF sequences Foreman and Erik, and on the SD sequence city, with the same coding parameters.
Figure 6.11 presents some PSNR comparisons between the side noisy de-scription and the central dede-scription obtained with the proposed algorithm of decoding of the central description, for Foreman. PSNR values are pre-sented for a bit-rate of 2 Mbps, and for different SNR values. Figures 6.12 and 6.13 present the same kind of results for the sequences Erik (Rt = 2 Mbps) and city (Rt = 2.5 Mbps). The gain in PSNR is here again im-portant: the performances in respect with the noisy side description can be improved up to 9 dB.
Visual results are also presented. Figure 6.14(a) shows, for a SNR equal at 3 dB, reconstructed images of Foreman, at Rt = 2 Mbps, for the noiseless images, the noisy side description and for the central description, obtained by applying the second approach. Figures 6.14(b) and 6.14(c) show the same results for Erik (SN R = 4 dB, Rt= 2 Mbps) and city (SN R = 4
6.2. Multiple description decoding 125
2 3 4 5 6 7 8
20 22 24 26 28 30 32 34 36
SNR (dB)
PSNR (dB)
Central (proposed) Side
Figure 6.11: Second approach: PSNR comparisons for Foreman between the side noisy description and the central description, bit-rate Rt= 2 Mbps.
2 3 4 5 6 7
20 22 24 26 28 30 32
SNR (dB)
PSNR (dB)
Central (proposed) Side
Figure 6.12: Second approach: PSNR comparisons for erik between the side noisy description and the central description, bit-rate Rt= 2 Mbps.
dB, Rt= 2.5 Mbps). Here again, the central descriptions (j) are better pre-served, thanks to the second approach.
In Table 6.2 are summarized the PSNR comparisons between the side
2 3 4 5 6 7 21
22 23 24 25 26 27 28 29
SNR (dB)
PSNR (dB)
Central (proposed) Side
Figure 6.13: Second approach: PSNR comparisons for city between the side noisy description and the central description, bit-rate Rt= 2.5 Mbps.
description and the central description obtained by applying the second approach of decoding, for different values of SN R, with the same coding parameters than previously for Foreman, Erik and city. The results are good, the PSNR of the central description is always higher than the one of the side description.
6.2.4 Comparison between the two approaches
Let’s do a brief comparison between the two proposed approaches of optimal decoding. Table 6.3 presents some results of central descriptions obtained with the two approaches, for the sequences Foreman, Erik and city, with the same coding parameters. The noiseless references are presented into parentheses. For low channel noises, the second approach gives better results than the first one. On the contrary, for higher noises, the first approach allows a much better reconstruction of the central description.
This behaviour can be explained by the fact that the first approach better describes the source and the MDC scheme, thanks to the joint density p(s1, s2). It can be however remarked that the first approach is more complex in terms of time computation, especially because of the calculus of the joint density of the two descriptions.
6.2. Multiple description decoding 127
(i) (j) (k)
(a) Foreman, SN R = 3 dB, Rt= 2 Mbps, images 13 and 117.
(b) erik, SN R = 4 dB, Rt= 2 Mbps, images 13 and 44.
(c) city, SN R = 4 dB, Rt= 2.5 Mbps, images 18 and 44.
Figure 6.14: Second approach: visual results, (i) noiseless images, (j) central description, (k) side description.
Foreman Foreman
1.5 Mbps Side Central 2 Mbps Side Central SN R = 3 20.67 29.13 SN R = 3 20.98 32.39 SN R = 7 23.74 34.00 SN R = 7 26.84 36.03
Erik Erik
1.5 Mbps Side Central 2 Mbps Side Central SN R = 2 19.03 20.59 SN R = 2 19.31 22.85 SN R = 6 24.15 31.74 SN R = 6 22.92 33.03
City City
2.5 Mbps Side Central 2.8 Mbps Side Central SN R = 4 23.7 25.94 SN R = 4 24.12 26.58 SN R = 7 25.68 28.58 SN R = 7 25.88 28.61
Table 6.2: Second approach: PSNR (dB) comparisons between the side description and the central description obtained with the first approach;
for the sequences Foreman, Erik and city (on three (2,0) decomposition levels, with quarter-pixel motion vectors), for different bit-rates, and with different values of SN R.
Foreman Foreman
1.5 Mbps Approach 1 Approach 2 2 Mbps Approach 1 Approach 2
(34.05) (36.07)
SN R = 3 30.34 29.13 SN R = 3 33.50 32.39
SN R = 7 33.72 34.00 SN R = 7 35.60 36.03
Erik Erik
1.5 Mbps Approach 1 Approach 2 2 Mbps Approach 1 Approach 2
(31.80) (33.10)
SN R = 2 28.23 20.59 SN R = 2 29.31 22.85
SN R = 6 31.32 31.74 SN R = 6 32.67 33.03
City City
2.5 Mbps Approach 1 Approach 2 2.8 Mbps Approach 1 Approach 2
(28.61) (28.67)
SN R = 4 27.68 25.94 SN R = 4 28.21 26.58
SN R = 7 28.28 28.58 SN R = 7 28.35 28.61
Table 6.3: PSNR (dB) comparisons for the central description between the two approaches of decoding (the noiseless references are presented into parentheses); for the sequences Foreman, Erik and city (on three (2,0) decomposition levels, with quarter-pixel motion vectors), for different bit-rates, and with different values of SN R.
6.2. Multiple description decoding 129
Figure 6.15: Performances comparison between the proposed approaches and the ML estimations, with the results in the basic case, for Foreman, at 2 Mbps.
6.2.5 Comparison with some other methods
In this section is presented a comparison, for Foreman (encoded using three temporal decomposition levels and a motion-compensation with quarter-pixelic motion vectors), between the two proposed decoding approaches, and some state-of-the-art methods.
Figure 6.15 presents curves for the PSNR of the obtained central de-scription using the proposed decoding algorithms, the Maximum Likelihood
Figure 6.15 presents curves for the PSNR of the obtained central de-scription using the proposed decoding algorithms, the Maximum Likelihood